Computer Science > Discrete Mathematics

Title:
Contagious Sets in Expanders

Abstract: We consider the following activation process in undirected graphs: a vertex
is active either if it belongs to a set of initially activated vertices or if
at some point it has at least $r$ active neighbors, where $r>1$ is the
activation threshold.
A \emph{contagious set} is a set whose activation results with the entire
graph being active. Given a graph $G$, let $m(G,r)$ be the minimal size of a
contagious set. Computing $m(G,r)$ is NP-hard.
It is known that for every $d$-regular or nearly $d$-regular graph on $n$
vertices, $m(G,r) \le O(\frac{nr}{d})$. We consider such graphs that
additionally have expansion properties, parameterized by the spectral gap
and/or the girth of the graphs.
The general flavor of our results is that sufficiently strong expansion
(e.g., $\lambda(G)=O(\sqrt{d})$, or girth $\Omega(\log \log d)$) implies that
$m(G,2) \le O(\frac{n}{d^2})$ (and more generally, $m(G,r) \le
O(\frac{n}{d^{r/(r-1)}})$). Significantly weaker expansion properties suffice
in order to imply that $m(G,2)\le O(\frac{n \log d}{d^2})$. For example, we
show this for graphs of girth at least~7, and for graphs with
$\lambda(G)<(1-\epsilon)d$, provided the graph has no 4-cycles. Nearly
$d$-regular expander graphs can be obtained by considering the binomial random
graph $G(n,p)$ with $p \simeq \frac{d}{n}$ and $d > \log n$. For such graphs we
prove that $\Omega(\frac{n}{d^2 \log d}) \le m(G,2) \le O(\frac{n\log\log
d}{d^2\log d})$ almost surely.
Our results are algorithmic, entailing simple and efficient algorithms for
selecting contagious sets.